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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.21245 |
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| _version_ | 1866912858019725312 |
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| author | Li, Yeping Wang, Gaofeng Wu, Tianfang |
| author_facet | Li, Yeping Wang, Gaofeng Wu, Tianfang |
| contents | In this paper, we study the hydrodynamic limit of the Vlasov-Poisson-Boltzmann system for a gas mixture in the whole space $(x \in \mathbb{R}^3)$ with the potential range of $γ\in\left(-3, 1\right]$. Using the method of Hilbert expansion, we first derive a bi-Maxwellian determined by the Euler-Poisson system of two fluids. To justify the convergence of the solution rigorously as the Knudsen number tends to zero, we sequentially calculate the first $2k-1$ terms of the expansion series $(k \geq 6)$, and then truncate it, and express the solution as the sum of these first $2k-1$ terms and a remainder term. Within the framework of the $L_{x,v}^2-W_{x,v}^{1,\infty}$ interplay established by Guo and Jang \cite{[ininp]Guo2010CMP}, we construct a new weight function to estimate the remainder term in four different cases regarding the potential $γ$. Here, the particle masses $m^A, m^B > 0$ and their charges $e^A, e^B$ can be given arbitrarily. This causes the collision operator to exhibit asymmetric effects ($m^A \neq m^B$), rendering the system of equations impossible to decouple. So, it adds difficulties to both $L^2$, $L^{\infty}$ estimates for the remainder. Therefore, we adopt the framework of vector-valued functions and analyze the velocity decay rate of the operator $K_{M,2,w}^{α,c}$ to eliminate the singularity induced by small parameters in characteristic line iterations. Our results show that the validity time of the solution is $O(\varepsilon^{-y})$, where $y$ is $-\frac{2k-3}{2(2k-1)}$ when $-1 \leq γ\leq 1$, and it becomes $-\frac{2k-3}{(1-γ)(2k-1)}$, when $-3 < γ< -1$. These results possess strong physical realism and can be applied to analyze gas flow dynamics in the daytime ionosphere at high altitudes above the Earth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_21245 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hydrodynamic limit of the Vlasov-Poisson-Boltzmann system for gas mixture Li, Yeping Wang, Gaofeng Wu, Tianfang Analysis of PDEs 35B38, 35J47 F.2.2 In this paper, we study the hydrodynamic limit of the Vlasov-Poisson-Boltzmann system for a gas mixture in the whole space $(x \in \mathbb{R}^3)$ with the potential range of $γ\in\left(-3, 1\right]$. Using the method of Hilbert expansion, we first derive a bi-Maxwellian determined by the Euler-Poisson system of two fluids. To justify the convergence of the solution rigorously as the Knudsen number tends to zero, we sequentially calculate the first $2k-1$ terms of the expansion series $(k \geq 6)$, and then truncate it, and express the solution as the sum of these first $2k-1$ terms and a remainder term. Within the framework of the $L_{x,v}^2-W_{x,v}^{1,\infty}$ interplay established by Guo and Jang \cite{[ininp]Guo2010CMP}, we construct a new weight function to estimate the remainder term in four different cases regarding the potential $γ$. Here, the particle masses $m^A, m^B > 0$ and their charges $e^A, e^B$ can be given arbitrarily. This causes the collision operator to exhibit asymmetric effects ($m^A \neq m^B$), rendering the system of equations impossible to decouple. So, it adds difficulties to both $L^2$, $L^{\infty}$ estimates for the remainder. Therefore, we adopt the framework of vector-valued functions and analyze the velocity decay rate of the operator $K_{M,2,w}^{α,c}$ to eliminate the singularity induced by small parameters in characteristic line iterations. Our results show that the validity time of the solution is $O(\varepsilon^{-y})$, where $y$ is $-\frac{2k-3}{2(2k-1)}$ when $-1 \leq γ\leq 1$, and it becomes $-\frac{2k-3}{(1-γ)(2k-1)}$, when $-3 < γ< -1$. These results possess strong physical realism and can be applied to analyze gas flow dynamics in the daytime ionosphere at high altitudes above the Earth. |
| title | Hydrodynamic limit of the Vlasov-Poisson-Boltzmann system for gas mixture |
| topic | Analysis of PDEs 35B38, 35J47 F.2.2 |
| url | https://arxiv.org/abs/2601.21245 |